The LMP2 Equations

In LMP2 fheory fhe MP2 equations are expressed using an orbital-invariant formulation employing noncanonical orbitals, and a number of approximations are introduced to achieve reduced scaling of the computational cost. 

The employed set of noncanonical orbitals are local in nature the occupied orbital space is represented by localized occupied molecular orbitals, and the virtual (unoccupied) orbital space consists of atomic orbitals (AOs) projected into the virtual orbital space to ensure the required orthogonality of the occupied and unoccupied spaces. 

In the orbital-invariant formulation, the closed-shell MP2 correlation energy can be expressed as follows  

The elements of the matrix Kij are the two-electron integrals, Kf- = ia jb), and the elements of the matrix Tjj are the double-substitution amplitudes Tj . Indices i, j and a, h represent occupied and unoccupied orbitals, respectively. 

The two-electron integrals ia jb) are computed by transformation of the AO integrals fuv kcr) in a four-step transformation similar to that used in conventional MP2 theory (cf. section 9.1) 

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